what is standard error of the mean?The chefs at a local pizza chain in Cambria California, strive to maintain the suggested size of their 16- inch pizzas. Despite their best efforts, they are unable to make every pizza exactly 16 inches in diameter. The manager had determined that the size of the pizzas is normally distributed with a mean of 16 inches and a population standard deviation of 0.8 inch.What are the expected value and the standard error of the sample mean derived from a random sample of 2 pizzas?What are the expected value and the standard error of the sample mean derived from a random sample of 4 pizzas?Compare the expected value and the standard error of the sample mean with those of an individual pizza.What is the probability that 2 randomly selected pizzas average less than 15.5 inches?What is the probability that 4 randomly selected pizzas average greater than 15.5 inches?

1. Consider that the mean and standard deviation of a random variable X are µ and σ, respectively. Thus, the z-score of that random variable X is z = (X – µ)/σ.Define a random variable X, that is the size of a pizza. According to the given information, X have follows normal distribution with mean µx = 16 inches and the standard deviation of σ x = 0.8 inches. Central limit theorem: If the population follows normal distribution or the sample size is large enough (>30), then the sampling distribution of the sample mean follows normal distribution with the expectation and standard deviation explained as follows: E(x̅) = µ and SD(x̅) = σ/√n. Here, for the random variable x the population mean is µ, the population standard deviation is σ, and the sample size is n. Denote that x̅ is the sample mean. Now, for sample of 2 pizzas, the mean of the sampling distribution of the sample mean is, µx̅ = 16. Now, for sample of 2 pizzas, the standard deviation of the sampling distribution of the sample mean is, σx̅ = 0.8/√2 = 0.565685424. Thus, the sampling distribution of x̅ is approximately normal with mean µx̅ = 16 and standard deviation σx̅ = 0.565685424. That is, the expected value and the standard error of the sample mean derived from a random sample of 2 pizzas are 16 inches and 0.565685424 inch. 2. Now, for sample of 4 pizzas, the mean of the sampling distribution of the sample mean is, µx̅ = 16. Now, for sample of 4 pizzas, the standard deviation of the sampling distribution of the sample mean is, σx̅ = 0.8/√4 = 0.4. Thus, the sampling distribution of x̅ is approximately normal with mean µx̅ = 16 and standard deviation σx̅ = 0.4. That is, the expected value and the standard error of the sample mean derived from a random sample of 4 pizzas are 16 inches and 0.4 inch. 3. It is clear that the expected value of the sample mean is same as the expected value of the individual pizza. However, the standard error of sample mean is less that the standard deviation of individual pizza. Moreover, as the sample size increases the standard error decreases. 4. The probability that 2 randomly selected pizzas average less than 15.5 inches is, Thus, the probability that 2 randomly selected pizzas average less than 15.5 inches is 0.1894. 5. The probability that 4 randomly selected pizzas average less than 15.5 inches is, Thus, the probability that 4 randomly selected pizzas average less than 15.5 inches is 0.1056.

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MATLAB: An Introduction with Applications

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ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

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what is standard error of the mean?

The chefs at a local pizza chain in Cambria California, strive to maintain the suggested size of their 16- inch pizzas. Despite their best efforts, they are unable to make every pizza exactly 16 inches in diameter. The manager had determined that the size of the pizzas is normally distributed with a mean of 16 inches and a population standard deviation of 0.8 inch.

What are the expected value and the standard error of the sample mean derived from a random sample of 2 pizzas?
What are the expected value and the standard error of the sample mean derived from a random sample of 4 pizzas?
Compare the expected value and the standard error of the sample mean with those of an individual pizza.
What is the probability that 2 randomly selected pizzas average less than 15.5 inches?
What is the probability that 4 randomly selected pizzas average greater than 15.5 inches?

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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